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In
addition to estimating the gradient of \(F\) , we also estimate a stochastic linear approximation of the inner functions \(f_i\)  by a moving-average technique. In the multi-level setting we consider, it helps us to avoid the accumulation of bias, when estimating the \(f_i\)  directly. Linearization techniques were used in the stochastic optimization since the work of [1]}. A similar approach was used in [2]} in the context of projected-based methods for solving (REF ). It is also worth mentioning that other linearization techniques have been used in [3]} and [4]} for estimating the stochastic inner function values for weakly convex two-level composition problems.

The ICG method given in Algorithm  is essentially applying deterministic conditional gradient method with the exact line search for solving the quadratic minimization subproblem in (REF ) with the estimated gradient \(z_k\)  in (REF ). It was also used in [5]} as a sub-routine and is motivated by the sliding approach of [6]}.


